In his fourth start since his trade from Cincinnati, Cueto went eight innings, scattered eight hits, gave up one run, walked none and struck out four.
"I feel really good on this ballclub," he said through interpreter Pedro Grifol. "Every time I go out there, I feel I have a chance to win because the offense gives me runs and gives me that boost."
The Royals also give their pitchers great defense. Catcher Salvador Perez picked a runner off first, and center fielder Jarrod Dyson gunned down David Murphy at the plate as Murphy tried to score from second on a single.
"Incredible defense," Cueto said. "It's an advantage for all of us pitchers to have that defense. For Sal to pick off that runner, that's a free out. It's an extra boost."
Saturday's game actually didn't start out all that well. In the first two innings, six Angels hitters hit the ball hard, including Albert Pujols, who belted a home run. Cueto was fortunate to escape only giving up one run.
"I was kind of aiming in the beginning in the first couple of innings," he said. "When Pujols hit the home run, it woke me up a little and I thought, 'It's time to roll.'"
And roll he did. Cueto shut out the Angels over his final six innings as the Royals kept adding to the lead. Cueto, though, said he didn't change his approach with the comfortable lead.
"Not at all," he said. "I continued to attack with all my pitches. Nothing changes."
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.